Rolling Excess Kurtosis (KURTOSIS)

Summary

An estimate, from the trailing window, of the excess kurtosis of the distribution its values come from: the fourth standardised moment minus 3, so a normal distribution reads 0. A tail-weight measure: above 0 the distribution has heavier tails than a normal one with the same variance, below 0 lighter ones. The companion to the shipped VAR and STDDEV in the same group.

Formula

Sample-adjusted Fisher excess kurtosis, the estimator usually written G2. With window size n, window mean x̄, and sample variance s² = Σ(x−x̄)²/(n−1):

G2 = [ n(n+1) / ((n−1)(n−2)(n−3)) ] · [ Σ(x−x̄)⁴ / s⁴ ]  −  3(n−1)² / ((n−2)(n−3))

The result is NaN when s² = 0, that is when every value in the window is equal.

optInTimePeriod must be at least 4: the (n−2)(n−3) denominators are undefined below it. The bound is enforced by the parameter range, so a shorter period is refused rather than silently degraded.

Notes

This is G2, the form Excel KURT and scipy.stats.kurtosis(bias=False) compute. The other form in circulation is the biased g2 = m₄/m₂² − 3 over population moments, a different estimator rather than a rounding convention: G2 = (n−1)((n+1)·g2 + 6) / ((n−2)(n−3)).
A point mass has no defensible excess kurtosis, so the degenerate window is NaN rather than a number: 0 would assert normality and −1.2 uniformity. This is deliberately unlike VAR, which floors to zero — a variance of zero says something true about the window.
The result is at most n, the value of one reading apart from n−1 equal ones, so a window dominated by one outlier is legitimately far above 0.

Inputs

inReal — The series to measure

Outputs

outReal — Excess kurtosis of the trailing window, or NaN where the window has no spread

Parameters

Parameter	Type	Default	Accepted values	Description
`optInTimePeriod`	integer	30	4–10000	Number of trailing values in the window

Properties

Numerical Stability: Start-Independent


☐ Overlap Input
✅ Independent Y-Axis i
☐ Candlestick
✅ Can Output NaN or ±Inf i
☐ Identity at Period 1

Implementation

TA-Lib Definition: kurtosis.c · kurtosis.yaml

Native	File
C	`ta_KURTOSIS.c`
Rust	`kurtosis.rs`
Java	`Core_KURTOSIS.java`
C#	`Core_KURTOSIS.cs`

TA-Lib is also available for Python, R and more using a wrapper.

Aliases

Excess Kurtosis, Sample Excess Kurtosis, KURT

References

Joanes, D. N. and Gill, C. A. "Comparing measures of sample skewness and kurtosis." Journal of the Royal Statistical Society: Series D, 47(1), 1998, 183-189 — the g1/g2, G1/G2, b1/b2 families and which is unbiased under which assumption.
NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.11 — the biased population form.
Microsoft, KURT function — the G2 form, and #DIV/0! for fewer than four points or zero standard deviation.